3D weak and galaxy clustering using spin wavelets on the ball

Talk given at the Cosmostats conference in Berkeley summarizing arXiv:1509.06749 and arXiv:1509.06750

Transcript

1. 3D weak lensing & galaxy clustering with spin wavelets on the ball Boris Leistedt Simons Fellow, New York University @ixkael - ixkael.com
2. Based on the recent works: 1205.0792, 1509.06749, 1509.06750 with Hiranya Peiris Jason McEwen Tom Kitching Martin Büttner Codes available at s2let.org (2D) & flaglets.org (3D)
3. accuracy (good methods) precision (good data)
4. ROAD MAP Galaxy surveys and data on the ball Fourier-Laguerre transform Flaglets and 3D cosmic shear
5. Cosmology with galaxy surveys Dark matter Dark energy GR / modified gravity Origin of structure Exotic physics Dark matter Galaxies
6. spectroscopic photometric credit: Aragon-Calvo et al (2014) z z 3D 3D + 2+1D types + redshifts no lensing shallow lensing deep no types / redshifts
7. State of the art 2+1D real or harmonic space analyses Fourier-Bessel (see e.g. Kitching et al papers) dealing w masks, modeling, covariances, systematics is difficult & intensive Address these issues using novel 3D transforms
8. Wish list for 3D transforms Separable, on rather than Fast & accurate, with 3D pixelization Relate to Fourier-Bessel B3 = S2 ⇥ R+ R3
9. The Fourier- Laguerre transform
10. Fourier-Laguerre basis: Separable basis on with measure d3~ r = r2 sin ✓d✓d dr S2 ⇥ R+ Leistedt & McEwen (2012) 3D with xz slices slice z = 0 slice x = 0 half sphere r = R/ 4 half sphere r = R/ 2 h Real part Imag. part sZ`mp(✓, , r) ⌘ sY`m(✓, ) Kp(r)
11. Leistedt & McEwen (2012) Band limited signal: harmonic coefficients L2P Exact when using sampling theorem (=pixelization + quadrature rule) sf(✓, , r) = L 1 X `=0 ` X m= ` P X p=0 sf`mp sY`m(✓, )Kp(r) sf`mp = ZZZ sf(✓, , r) sY ⇤ `m (✓, ) K⇤ p (r) r2 sin ✓d✓d dr
12. 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 0 0.2 0.4 0.6 0.8 1 −0.02 −0.01 0 0.01 0.02 r r Kp (r) r rKp (r) reconstructed on calculated from } samples ⇠ 2PL2 f`mp f
13. Connection to Fourier-Bessel analysis Fourier-Bessel transform: Hard to compute since no sampling theorem but linear sum of Fourier-Laguerre coefficients => Exact Fourier-Bessel transform for generic signals Leistedt & McEwen (2012) sf`m(k) = ZZZ sf(✓, , r) sY ⇤ `m (✓, ) j⇤ ` (kr) r2 sin ✓d✓d dr
14. Cosmology with Fourier-Laguerre ‣ Fast, exact transform (pixelization, FFTs, recurrences) ‣ Direct relation to Fourier-Bessel ‣ Easy to deconvolve mask ‣ Better control over scales probed ‣ Spatial + radial systematics marginalization
15. Fourier-Laguerre wavelets aka Flaglets
16. Flaglet transform = convolution Construct flaglets to: ‣ probe well defined radial & angular scales ‣ probe N complementary orientations about ‣ achieve exact reconstruction & multi resolution in harmonic space thanks to sampling theorem Leistedt & McEwen (IEEE, 2012) (✓, ) sWs ij (✓, , , r) ⌘ sf ? s ij = hsf|R(✓, , )s iji s ij Leistedt, McEwen, Kitching & Peiris (PRD, 2015) hsf|R(✓, , ) Trs iji
17. Axisymmetric flaglets smaller angular aperture (i↑) smaller radial extent (j↑) ij(✓, , r)
18. Flaglet transform of Horizon simulation Input signal Scaling fct Flaglet: i=0,j=0 Flaglet: i=0,j=1 Flaglet: i=1,j=0 Flaglet: i=1,j=1
19. Axisymmetric scalar wavelets Directional scalar wavelets Directional spin 2 wavelets real part imag part
20. scalar & spin directional flaglets
21. Spin s = 0 flaglets with = ⌫ = 3, I0 = J0 = 2 and N = 3 N = 2 N = 1 3D with xz slices slice z = 0 slice x = 0 half sphere r = R/ 2 half sphere r = R/ 2 half sphere r = R/ 2 i = 2 j = 2 i = 2 j = 3 i = 3 j = 2 i = 3 j = 3 Leistedt, McEwen, Kitching & Peiris (PRD, 2015)
22. Real part of spin s = 2 flaglets with = ⌫ = 3 , I0 = J0 = 2 , N = 1 Imag. part Modulus 3D with xz slices slice z = 0 slice x = 0 half sphere r = R/ 2 half sphere r = R/ 2 half sphere r = R/ 2 i = 2 j = 2 i = 2 j = 3 i = 3 j = 2 i = 3 j = 3 Leistedt, McEwen, Kitching & Peiris (PRD, 2015)
23. 3D cosmic shear with flaglets
24. 3D Cosmic shear with flaglets Flaglet transform: Covariance of flaglet coefficients: Leistedt, McEwen, Kitching & Peiris (PRD, 2015) Easy to calculate from theory and data sW ij (✓, , r) = (2 ? 2 ij)(✓, , r) h 2W ij (✓, , r) 2W i0j0⇤ (✓0, 0, r0) i = 2 ⇡ X ` (N`,2)2 4 ⇥ Z dk k2 Z dk0 k02 C ` P`( ✓) 2 Hij ` (k, r) 2 Hi0j0⇤ ` (k0, r0) = Cij,i0j0 (cos ✓, r, r0 )
25. Fast flaglet estimator (on data & theory) Can deal with complicated masks Can cut out unreliable modes Extendable to galaxy clustering, shear, flexion, etc Next: redshift uncertainties & spatial systematics Leistedt, McEwen, Kitching & Peiris (PRD, 2015)
26. conclusions
27. 3D cosmology with novel transforms [Leistedt+2012, 2015, flaglets.org, s2let.org] Coming soon: application to data Related: CMB map-making (Keir’s talk, 1601:01322) E-B separation (Leistedt+, in prep) 3D Data Compression LSST taskforce Summary